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Question Number 121502 by bramlexs22 last updated on 08/Nov/20

$$\left(1\right)\underset{x\to 0}{\lim }\frac{\sqrt{1-\cos 2\mathrm{x}}}{\mathrm{x}\sqrt{2}}?$$$$\left(2\right)\int \cot {}^3\mathrm{x}\csc {}^3\mathrm{x}\mathrm{dx}$$$$$$

Commented by liberty last updated on 09/Nov/20

$$\left(2\right)\int \cot {}^2\mathrm{x}{\csc }^2\mathrm{x}\left(\cot \mathrm{x}\csc \mathrm{x}\right)\mathrm{dx}=$$$$\iff \int {\csc }^2\mathrm{x}({\csc }^2\mathrm{x}-1)\left(\cot \mathrm{x}\csc \mathrm{x}\right)\mathrm{dx}=$$$$\iff \int {\csc }^2\mathrm{x}(1-{\csc }^2\mathrm{x})\mathrm{d}\left(\csc \mathrm{x}\right)=$$$$\iff \int ({\mathrm{z}}^2-{\mathrm{z}}^4)\mathrm{dz}=\frac{1}{3}{\mathrm{z}}^3-\frac{1}{5}{\mathrm{z}}^5+\mathrm{c}$$$$\iff \frac{1}{3}{\csc }^3\mathrm{z}-\frac{1}{5}{\csc }^5\mathrm{z}+\mathrm{c}$$$$\iff \frac{1}{3\sin {}^3\mathrm{x}}-\frac{1}{5\sin {}^5\mathrm{x}}+\mathrm{c}$$

Commented by liberty last updated on 09/Nov/20

$$\left(1\right)\underset{x\to 0}{\lim }\frac{\sqrt{1-\cos 2\mathrm{x}}}{\mathrm{x}\sqrt{2}}=\underset{x\to 0}{\lim }\frac{\sqrt{2}\sqrt{\sin {}^2\mathrm{x}}}{\mathrm{x}\sqrt{2}}$$$$\underset{x\to 0}{\lim }\frac{\mid \sin \mathrm{x}\mid }{\mathrm{x}}\to \{\begin{array}{l}\underset{x\to 0^{+}}{\lim }\frac{\mid \sin \mathrm{x}\mid }{\mathrm{x}}=\underset{x\to 0^{+}}{\lim }\frac{\sin \mathrm{x}}{\mathrm{x}}=1\\ \underset{x\to 0^{-}}{\lim }\frac{\mid \sin \mathrm{x}\mid }{\mathrm{x}}=\underset{x\to 0^{-}}{\lim }\frac{-\sin \mathrm{x}}{\mathrm{x}}=-1\end{array}$$$$\mathrm{then}\underset{x\to 0}{\lim }\frac{\sqrt{1-\cos 2\mathrm{x}}}{\mathrm{x}\sqrt{2}}\mathrm{doesnot}\mathrm{exsist}$$

Answered by Lordose last updated on 08/Nov/20

$$1.\underset{x\to 0}{\lim }\frac{\sqrt{{\cos }^2\mathrm{x}+{\sin }^2\mathrm{x}-{\cos }^2\mathrm{x}+{\sin }^2\mathrm{x}}}{\mathrm{x}\sqrt{2}}$$$$\underset{x\to 0}{\lim }\frac{\sqrt{2}\mathrm{sinx}}{\mathrm{x}\sqrt{2}}=\underset{x\to 0}{\lim }\frac{\mathrm{sinx}}{\mathrm{x}}=1$$

Commented by liberty last updated on 12/Nov/20

$$\mathrm{wrong}$$

Answered by Lordose last updated on 08/Nov/20

$$2.\int \cot \left(\mathrm{x}\right){\csc }^3\mathrm{x}({\csc }^2\mathrm{x}-1)\mathrm{dx}\{{\cot }^2\mathrm{x}={\csc }^2\mathrm{x}-1\}$$$$\int ({\mathrm{u}}^2-{\mathrm{u}}^4)\mathrm{du}\{\mathrm{u}=\csc \left(\mathrm{x}\right)\Rightarrow \mathrm{du}=-\mathrm{ucotxdx}\}$$$$\frac{{\mathrm{u}}^3}{3}-\frac{{\mathrm{u}}^5}{5}+\mathrm{C}$$$$\frac{1}{15}({5\csc }^3\mathrm{x}-{3\csc }^5\mathrm{x})+\mathrm{C}$$

Answered by Bird last updated on 09/Nov/20

$$wehavecos\left(2x\right)\sim 1-2x^2\Rightarrow$$$$1-cos\left(2x\right)\sim 2x^2\Rightarrow \sqrt{1-cos\left(2x\right)}\sim \sqrt{2}\mid x\mid$$$$\Rightarrow \frac{\sqrt{1-cos\left(2x\right)}}{x\sqrt{2}}\sim \frac{\sqrt{2}\mid x\mid }{x\sqrt{2}}=f\left(x\right)\Rightarrow$$$${lim}_{x\to 0^{+}}f\left(x\right)=1and$$$${lim}_{x\to 0^{-}}f\left(x\right)=-1$$

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